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MathidiaMathematics Academy
Linear systems and matrices
Lesson 20 of 75
Review

Linear system models

Objective. Form and critique systems arising from applications.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Linear system models translate real situations — mixtures, networks, economics, circuits — into equations where unknowns appear only to the first power, then represent the resulting equations as a matrix system Ax=b to be solved and interpreted in context. Critiquing a model means checking that the equations correctly capture each stated constraint (e.g. total cost, conservation of flow) and that the solution makes sense, such as nonnegative quantities when modelling physical amounts. For a mixture problem needing x litres of 10% acid and y litres of 30% acid to make 10 litres of 18% acid, the equations are x+y=10 and 0.10x+0.30y=1.8, solved to give x=6, y=4. A model producing a negative or non-integer answer where only whole positive quantities make sense should be revisited.

Worked example

Form and solve the linear system model for mixing x litres of 10% acid and y litres of 30% acid to obtain 10 litres of 18% acid.

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Bar chart of the values 10, 30, 10, 18
  1. 1Volume equation: x+y=10, since the two amounts combine to the total volume.
  2. 2Acid-content equation: 0.10x+0.30y=1.8, matching the required 18% of 10 litres.
  3. 3From the first equation, y=10−x; substitute into the second: 0.10x+0.30(10−x)=1.8, giving 3−0.20x=1.8.
  4. 4Solve: 0.20x=1.2, so x=6, and y=10−6=4.

Answer: x = 6 litres, y = 4 litres

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Common mistake. Students set up a percentage mixture equation using whole percentages instead of decimals, for example writing 10x+30y=18 instead of 0.10x+0.30y=1.8, which misscales the equation relative to the total-volume equation.

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